Linear-Time $(1+\varepsilon)$-Approximation Algorithms for Two-Line-Center Problems

Abstract

Given a set $S$ of $n$ points in the plane, we study the two-line-center problem: finding two lines that minimize the maximum distance from each point in $S$ to its closest line. We present a $(1+\varepsilon)$-approximation algorithm for the two-line-center problem that runs in $O((n/\varepsilon) \log (1/\varepsilon))$ time, which improves the previously best $O(n\log n + ({n}/{\varepsilon^2}) \log ({1}/{\varepsilon}) + (1/\varepsilon^3)\log ({1}/{\varepsilon}))$-time algorithm. We also consider three variants of this problem, in which the orientations of the two lines are restricted: (1) the orientation of one of the two lines is fixed, (2) the orientations of both lines are fixed, and (3) the two lines are required to be parallel. For each of these three variants, we give the first $(1+\varepsilon)$-approximation algorithm that runs in linear time. In particular, for the variant where the orientation of one of the two lines is fixed, we also give an improved exact algorithm that runs in $O(n \log n)$ time and show that it is optimal.

Figures (23)

• Figure 1: In each case, both yellow and gray pairs satisfy the restriction, but the yellow pair has the smaller maximum width.
• Figure 2: (a) The orientation of $\sigma$ is $\alpha$. (b) For a set $P$ of points and an orientation $\theta$, $\sigma_{\theta}(P)$ and $\mathrm{width}_{\theta}(P)$ are shown. Observe that $(p, q)$ is an antipodal pair of $P$ with respect to $\theta$.
• Figure 3: (a) Both $(p_2, p_3)$ and $(p_3, p_4)$ are anchor pairs of $\Sigma$, satisfying (1) and (2), respectively. (b) $(p_1', p_2')$ is an anchor pair of $\Sigma'$, satisfying (1), but $(p_4', p_6')$ is not.
• Figure 4: (a) There exists a point $r \in S \setminus (D_p \cup D_q)$. (b) The inner common tangents $\ell_1$ and $\ell_2$ of ${conv}(P)$ and ${conv}(Q)$, and the points $p_1, p_2, q_1, q_2$, and $s$ are shown. (c) The triangle $\triangle p_1 p_2 q'$ and the segment $\overline{q_1 q_2}$ intersect. (d) The segments $\overline{p_1 p_2}$ and $\overline{q_1 q_2}$ intersect.
• Figure 5: For three points $\alpha, \beta, \gamma \in \mathbb R^2$, the gray region represents $\sigma(\alpha, \beta ;\gamma)$.

Theorems & Definitions (47)

• Definition 1: Anchor pair
• Lemma 2
• Lemma 3
• Theorem 4: Theorem A.3 of chan2006faster
• Theorem 5
• Definition 6: $\varepsilon$-expansion
• Definition 7: $\varepsilon$-certificate
• Theorem 8
• Theorem 9
• Lemma 10: ahn2024constrained